Data compression systems are useful to represent information as accurately as possible with a minimum number of bits and thus minimize the amount of data which must be stored or transmitted in an information storage or transmission system. One of the primary means of doing this is to remove redundant information From the original data. In the Proceedings of the International Conference on Acoustics, Speech and Signal Processing, San Francisco, Cal. March 1992, volume IV, pages 657-660, I disclosed a signal compression system which applies a hierarchical subband decomposition, or wavelet transform, followed by the hierarchical successive approximation entropy-coded quantizer incorporating zerotrees. The representation of signal data using a multiresolution hierarchical subband representation was disclosed by Burr et al in IEEE Trans. on Commun. Vol Com-31, No 4. April 1983, page 533. A wavelet pyramid, or a critically sampled quadrature-mirror filter (QMF) subband representation is a specific kind of multiresolution hierarchical subband representation. A wavelet pyramid was disclosed by Pentland et al in Proc. Data Compression Conference Apr. 8-11, 1991, Snowbird, Utah. A QMF subband pyramid has been described in "Subband Image Coding", J. W. Woods ed., Kluwer Academic Publishers, 1991 and I. Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics (SIAM): Philadelphia, Pa., 1992.
Wavelet transforms otherwise known as hierarchical subband decomposition have recently been used for low bit rate image compression because it leads to a hierarchical multi-scale representation of the source image. Wavelet transforms are applied to an important aspect of low bit rate image coding: the coding of the binary map indicating the locations of the non-zero values, otherwise known as the significance map. Using scalar quantization followed by entropy coding, in order to achieve very low bit rates, i.e., less than 1 bit/pel, the probability of the most likely symbol after quantization--the zero symbol--must be extremely high. Typically, a large fraction of the bit budget must be spent on encoding the significance map. It follows that a significant improvement in encoding the significance map translates into a significant improvement in the compression of information preparatory to storage or transmission.
To accomplish this task, a new data structure called a zerotree is defined. A wavelet coefficient is said to be insignificant with respect to a given threshold T if the coefficient has a magnitude less than or equal to T. The zerotree is based on the hypothesis that if a wavelet coefficient at a coarse scale is insignificant with respect to a given threshold T, then all wavelet coefficients of the same orientation in the same spatial location at finer scales are likely to be insignificant with respect to T. Empirical evidence suggests that this hypothesis is often true.
More specifically, in a hierarchical subband system, with the exception of the highest frequency subbands, every coefficient at a given scale can be related to a set of coefficients at the next finer scale of similar orientation. The coefficient at the coarsest scale will be called the parent node, and all coefficients corresponding to the same spatial or temporal location at the next finer scale of similar orientation will be called child nodes. For a given parent node, the set of all coefficients at all finer scales of similar orientation corresponding to the same location are called descendants. Similarly, for a given child node, the set of coefficients at all coarser scales of similar orientation corresponding to the same location are called ancestors. With the exception of the lowest frequency subband, all parent nodes have four child nodes. For the lowest frequency subband, the parent-child relationship is defined such that each parent node has three child nodes.
A scanning of the coefficients is performed in such a way that no child node is scanned before any of its parent nodes. Given a threshold level to determine whether or not a coefficient is significant, a node is said to be a ZEROTREE ROOT if 1) the coefficient has an insignificant magnitude, 2) the node is not the descendant of a root, i.e. it is not completely predictable from a coarser scale, and 3) all of its descendants are insignificant. A ZEROTREE ROOT is encoded with a special symbol indicating that the insignificance of the coefficients at finer scales is completely predictable. To efficiently encode the binary significance map, three symbols are entropy coded: ZEROTREES, ISOLATED ZEROS, and non-zeros.